by Selim Bora

1
Business Research Methods

• Lecture 4-5
• by Selim Bora

2
The Idea of Probability

• Chance behavior is unpredictable in the short run
  but has a regular and predictable pattern in the
  long run.
• We call a phenomenon random if individual
  outcomes are uncertain but there is nonetheless a
  regular distribution of outcomes in a large
  number of repetitions.
• The probability of any outcome of a random
  phenomenon is a number between 0(the outcome
  never occurs) and 1(always occurs) that describes
  the proportion of times the outcome would occur
  in a very long series of repetitions.
• Probabilities describe only what happens in the
  long run, short runs of random phenomena often
  dont look random to us because they do not show
  the regularity that in fact emerges only in very
  many repetitions.

3
Personal Probabilities

• A personal probability of an outcome is a number
  between 0 and 1 that expresses an individuals
  judgment of how likely the outcome is.
• Different people can have different personal
  probabilities, and a personal probability need
  not agree with a proportion based on data about
  similar cases.

4
Probability Models

• A probability model for a random phenomenon
  describes all the possible outcomes and says how
  to assign probabilities to any collection of
  outcomes. We sometimes call a collection of
  outcomes an event.
• There are two simple ways to give a probability
  model. The first assigns a probability to each
  individual outcome. These probabilities must be
  numbers between 0 and 1, and they must add to
  exactly 1.
• The second kind of probability model assigns
  probabilities as areas under a density curve.

5
Probability Rules

• Any probability is a number between 0 and 1.
• All possible outcomes together must have
  probability 1.
• The probability that an event does not occur is 1
  minus the probability that the event does not
  occur.
• If two events have no outcomes in common, the
  probability that one or the other occurs is the
  sum of their individual probabilities.
• Any assignment of probabilities to all individual
  outcomes that satisfies first two rules is
  legitimate.

6
Sampling Distribution

• The sampling distribution of a statistic tells us
  what values the statistic takes in repeated
  samples from the same population and how often it
  takes those values.
• We think of a sampling distribution as assigning
  probabilities to the values the statistic can
  take. Because there are usually many possible
  values, sampling distributions are often
  described by a density curve.
• The total probability is 1 because the total area
  under the curve is 1.

7
Simulation

• Using random digits from a table or from computer
  software to imitate chance behavior is called
  simulation.
• We can use random digits to simulate random
  outcomes if we know the probabilities of the
  outcomes in three stages
• Give probability model
• Assign digits to represent outcomes
• Simulate many repetitions
• Two random phenomena are independent if knowing
  the outcome of one does not change the
  probabilities for outcomes of the other.

8
More Elaborate Simulations

• Other simulations may require varying number of
  trials or different probabilities at each stage
  or may have stages that are not independent so
  that the probabilities at some stage depend on
  the outcome of earlier stages.
• A tree diagram can be helpful by giving the
  probability model in graphical form.

9
Expected Values

• When the outcomes are numbers, as in games of
  chance, we are also interested in the long-run
  average outcome.
• The expected value of a random phenomenon that
  has numerical outcomes is found by multiplying
  each outcome by its probability and then adding
  all the products.
• In symbols, if the possible outcomes are a1, a2,
  , ak and their probabilities are p1, p2, , pk,
  the expected value is
• expected value a1p1 a2p2 akpk

10
The Law of Large Numbers

• According to the law of large numbers, if a
  random phenomenon with numerical outcomes is
  repeated many times independently, the mean of
  the actually observed outcomes approaches the
  expected value.
• If you dont know the outcome probabilities, you
  can estimate the expected value(along with
  probabilities) by simulation.